Research Interests:Mathematical modeling in biology and pattern formation

Bifurcation analysis of a model of hepatitis B

Hepatitis B is a virus that attacks the liver. In most cases, the immune system is able to fight off the virus; however, in some cases the immune system is not able to do this and a chronic infection results. We are conducting a bifurcation analysis of a known model for the spread of the hepatitis B virus in the body. Based on the results of this analysis, we hope to better understand which aspects of the immune response are most important to preventing a chronic infection.

Pattern selection in reaction-diffusion equations near a Turing-Hopf bifurcation.

Alan Turing proposed in the 1950s that a chemical reaction with a rapidly diffusing activator and a slowly diffusing inhibitor could exhibit an instability to a patterned state with a predictable wavenumber. After decades, his intuition has been shown to be true. Reaction-diffusion systems are also known to exhibit a Hopf instability, an instability to a state which is spatially homogeneous but oscillates temporally with a predictable frequency. Recent experiments have looked at the effect of blinking light on a chemical reaction which is exhibiting a Turing pattern. They found that the blinking light could suppress the pattern and that the suppression was greatest at the Hopf frequency of the system. Using perturbation theory and symmetry-based nonlinear analysis, my collaborator and I are looking at the behavior of general reaction-diffusion equations near the codimension-2 point where Turing and Hopf bifurcations occur simulaneously. The analysis will provide us with the tools to determine how external forcing (in the case of the aforementioned experiment, this would be the blinking light) could be used to contral the patterns seen in the reaction-diffusion system. Using a simulation we wrote of a the Lengyel-Epstein reaction-diffusion system, we can test these results numerically.

Mathematical modeling in neuroscience

Neurons and glial cells are two kinds of cells in the brain. Neurons communicate via rapid firing; glial cells communicate on a slower time scale via diffusion. Recently it has been shown that these two types of cells can also communicate with one another; however it is not clear what role neuro-glia communcation plays in neural networks. A group of glial biologists have suggested that one effect is to enhance the activity of neurons near excited glial cells and to suppress the activity of neurons far from excited glial cells. Their suggestion was based on intuition about the chemical reactions which occur near excited glial cells. I am part of a collaboration that has developed a simple reaction-diffusion model to describe these reactions and are using it to explore the biologists' suggestion via simulation and analysis of parameter space.